The distance formula on the plane follows from the Pythagorean theorem.

which can be viewed as a version of the Pythagorean theorem.

Image: Pythagorean. svg | Pythagoras ' theorem: The sum of the areas of the two squares on the legs ( a and b ) of a right triangle equals the area of the square on the hypotenuse ( c ).

The celebrated Pythagorean theorem ( book I, proposition 47 ) states that in any right triangle, the area of the square whose side is the hypotenuse ( the side opposite the right angle ) is equal to the sum of the areas of the squares whose sides are the two legs ( the two sides that meet at a right angle ).

In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms.

In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

* Pythagorean theorem

It can be viewed as a form of the Pythagorean theorem.

Animation illustrating Pythagorean theorem | Pythagoras ' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.

For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras ; the Egyptians had a correct formula for the volume of a frustum of a square pyramid ;

The Pythagorean theorem was also known to the Babylonians.

Pythagorean theorem: a < sup > 2 </ sup > + b < sup > 2 </ sup > = c < sup > 2 </ sup >

Baudhayana ( c. 8th century BCE ) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as:,,,, and as well as a statement of the Pythagorean theorem for the sides of a square: " The rope which is stretched across the diagonal of a square produces an area double the size of the original square.

" It also contains the general statement of the Pythagorean theorem ( for the sides of a rectangle ): " The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together.

All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

# REDIRECT Pythagorean theorem

The proof relies on the Pythagorean theorem.

It is known from the Pythagorean theorem that a < sup > 2 </ sup > + b < sup > 2 </ sup > = c < sup > 2 </ sup >.

The latter was a result of the Pythagorean theorem and the former the assumption that a + b ≤ c. The contradiction means that it is impossible for both to be true and it is known that the Pythagorean theorem holds.

At any given x, a right-angled triangle connects x, y and r to the origin, hence it follows from the Pythagorean theorem that:

The Platonist seemed to outweigh the Aristotelian in Alan, but he felt strongly that the divine is all intelligibility and argued this notion through much Aristotelian logic combined with Pythagorean mathematics.

In those oblique coordinate systems the computations of distances and angles is more complicated than in standard Cartesian systems, and many standard formulas ( such as the Pythagorean formula for the distance ) do not hold.

1850 BCE " contains fifteen Pythagorean triples with quite large entries, including ( 13500, 12709, 18541 ) which is a primitive triple ,< ref > Three positive integers form a primitive Pythagorean triple if and if the highest common factor of is 1.

The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.

In mathematics, a generalized mean, also known as power mean or Hölder mean ( named after Otto Hölder ), is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.

As noted above this relationship between the three Pythagorean means is not limited to n equals 1 or 2 ; there is a relationship for all n. However, for n

A geometric construction of the three Pythagorean means of two numbers, a and b. Harmonic mean is denoted by H in purple color.

This study of numerology is based on the evidence of significant double-digit numbers in the Kabbalah, the I-ching, the Pythagorean numerology, the Tarot Arcana of the Eastern faiths, and the Runes of the Viking age.

Euclid IX 21 — 34 is very probably Pythagorean ; it is very simple material (" odd times even is even ", " if an odd number measures divides an even number, then it also measures divides half of it "), but it is all that is needed to prove that

This is called Pythagorean tuning because it was first discovered by Pythagoras.

The quadrivium is implicit in early Pythagorean writings and in the De nuptiis of Martianus Capella, although the term " quadrivium " was not used until Boethius early in the sixth century.

The length of the resulting segment is the geometric mean, which can be proved using the Pythagorean theorem.

The implications of the above statement are profound because it is directly translated into Pythagorean Theorem ( and graphically represented in the picutre on the left ) and it becomes evident that Baudhāyana proved Pythagoras theorem.

Alcott described his sustenance as a " Pythagorean diet ": meat, eggs, butter, cheese, and milk were excluded and drinking was confined to well water.

* Pythagorean expectation: estimates a team's expected winning percentage based on runs scored and runs allowed.

Zopyrus has been plausibly equated with a Pythagorean of that name who seems to have flourished in the late 5th century BC.

Zopyrus has been plausibly equated with a Pythagorean of that name who seems to have flourished in the late 5th century BC.

* Manas, John Helen, Divination, ancient and modern, New York, Pythagorean Society, 1947.

< td > For n = 2 there are infinitely many solutions ( x, y, z ): the Pythagorean triples.

According to, the Śulba Sūtras contain " the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians.

They contain lists of Pythagorean triples ,< ref > Pythagorean triples are triples of integers with the property:.